Some refined results on mixed Littlewood conjecture for pseudo-absolute values

TL;DR

This paper refines the understanding of the mixed Littlewood conjecture involving pseudo-absolute values, providing sharp criteria for the existence of infinitely many coprime solutions to certain inequalities for almost every real number.

Contribution

It establishes sharp conditions under which the mixed Littlewood conjecture holds for pseudo-absolute values and multiple sequences, extending previous results.

Findings

Sharp criterion for pseudo-absolute value sequences

Almost sure existence of coprime solutions for inequalities

Extension to multiple pseudo-absolute value sequences

Abstract

In this paper, we study the mixed Littlewood conjecture with pseudo-absolute values. For any pseudo absolute value sequence $\mathcal{D}$, we obtain the sharp criterion such that for almost every $\alpha$ the inequality \begin{equation*} |n|_{\mathcal{D}}|n\alpha -p|\leq \psi(n) \end{equation*} has infinitely many coprime solutions $(n,p)\in\N\times \Z$ for a certain one-parameter family of $\psi$. Also under minor condition on pseudo absolute value sequences $\mathcal{D}_1$,$\mathcal{D}_2,\cdots, \mathcal{D}_k$, we obtain a sharp criterion on general sequence $\psi(n)$ such that for almost every $\alpha$ the inequality \begin{equation*} |n|_{\mathcal{D}_1}|n|_{\mathcal{D}_2}\cdots |n|_{\mathcal{D}_k}|n\alpha-p|\leq \psi(n) \end{equation*} has infinitely many coprime solutions $(n,p)\in\N\times \Z$.